about me

I am an assistant professor in the Department of Mathematics at Union College. Previously, I was a Hans Rademacher Instructor of Mathematics at the University of Pennsylvania. I received my Ph.D. in 2010 from Duke University under the direction of Hubert Bray. Prior to that, I received my bachelor's degree from Harvey Mudd College. Here is my CV.

teaching

  • Spring 2018: Math 130 (Ordinary Differential Equations)
  • Winter 2018: Integrated Math/Physics 120, Math 140 (Applied Linear Algebra)
  • (see CV for complete list)
    Here are archives of my course webpages from UPenn.

  • research

    primary interests: geometric analysis, mathematical relativity

    I work in geometric analysis, emphasizing connections with general relativity. Einstein's theory of general relativity describes the universe as a spacetime, which is a four-dimensional continuum containing all points and events, past, present and future. Gravitational effects (for instance, due to a black hole) manifest through the curvature of spacetime, and thus geometry plays an important role in the theory. My research typically involves scalar curvature and explores connections between mass and geometry, including "quasi-local" mass and the total "ADM" mass of a spacetime. My current interests include convergence of sequences of asymptotically flat manifolds, Bartnik's quasi-local mass conjectures, and codimension-two geometric flows within a spacetime.

    papers

    Note: links below are to preprints, not final versions.

  • Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass, (2018).
  • Lower semicontinuity of the ADM mass in dimensions two through seven, (2018).
  • (with M. Anderson) Embeddings, immersions and the Bartnik quasi-local mass conjectures, (2016).
  • (with D. Lee) Lower semicontinuity of mass under C0 convergence and Huisken's isoperimetric mass, Journal für die reine und angewandte Mathemtik (Crelle's Journal), to appear.
  • On the lower semicontinuity of the ADM mass, Communications in Analysis and Geometry, to appear.
  • (with H. Bray and M. Mars) Time flat surfaces and the monotonicity of the spacetime Hawking mass II, Annales Henri Poincaré, Vol. 17, No. 6, pp. 1457-1475.
  • (with H. Bray) On curves with nonnegative torsion, Archiv der Mathematik., Vol. 104, No. 6 (2015), pp. 561-575.
  • (with W. Wylie) Conformal diffeomorphisms of gradient Ricci solitons and generalized quasi-Einstein manifolds, Journal of Geometric Analysis, Vol. 25, No. 1 (2015), pp. 668-708.
  • (with H. Bray) Time flat surfaces and the monotonicity of the spacetime Hawking mass, Communications in Mathematical Physics, Vol. 335, No. 1 (2015), pp. 285-307.
  • (with H. Li**, C. Fenton, C. Chee, A.G.C. Bergqvist) Epilepsy Treatment Simplified through Mobile Ketogenic Diet Planning, Journal of Mobile Technology in Medicine, Vol. 3, No. 2 (2014).
  • (with P. Miao and L.-F. Tam) Extensions and fill-ins with nonnegative scalar curvature,
    Classical and Quantum Gravity 30 (2013) 195007. Chosen for inclusion in IOP Select.
  • Fill-ins of nonnegative scalar curvature, static metrics, and quasi-local mass,
    Pacific Journal of Mathematics, Vol. 261, No. 2 (2013), pp. 417-444.
  • Invariants of the harmonic conformal class of an asymptotically flat manifold,
    Communications in Analysis and Geometry, Vol. 20, No. 1 (2012), pp. 163-202.
  • (with H. Bray) A geometric theory of zero area singularities in general relativity,
    Asian Journal of Mathematics, Vol. 17 No. 3 (2013).
  • Penrose-type inequalities with a Euclidean background, Annals of Global Analysis and Geometry, to appear.
  • (thesis) Mass estimates, conformal techniques, and singularities in general relativity
  • **denotes undergraduate coauthor

    mathematical reviews

    Here is a link to review articles I have written for Mathematical Reviews. (MathSciNet access required)